
from matplotlib_venn import venn2, venn2_circles, venn2_unweighted
from matplotlib_venn import venn3, venn3_circles
from matplotlib import pyplot as plt
%matplotlib inline


def drawVenn2(A,B,labels = ('A', 'B')):
    '''
    listing elements in Venn diagram of two sets
    '''
    v2=venn2([A,B],set_labels = labels);
    v2.get_label_by_id('01').set_text(', '.join(map(str,B-A)))
    v2.get_label_by_id('10').set_text(', '.join(map(str,A-B)))
    if A&B:
        v2.get_label_by_id('11').set_text(', '.join(map(str,A&B)))


def drawVenn3(A,B,C,labels = ('A', 'B', 'C')):
    '''
    listing elements in Venn diagram of three sets, A, B, C
    '''
    v3=venn3([A,B,C],set_labels = labels);
    if A-B-C:
        v3.get_label_by_id('100').set_text('\n'.join(map(str,A-B-C)))
        v3.get_patch_by_id('100').set_alpha(0.6)
        v3.get_patch_by_id('100').set_facecolor('Yellow')
    if A&B-C:
        v3.get_label_by_id('110').set_text('\n'.join(map(str,A&B-C)))
        v3.get_patch_by_id('110').set_alpha(0.3)
        v3.get_patch_by_id('110').set_facecolor('Red')
    if B-C-A:
        v3.get_label_by_id('010').set_text('\n'.join(map(str,B-C-A)))
        v3.get_patch_by_id('010').set_alpha(0.6)
        v3.get_patch_by_id('010').set_facecolor('Magenta')
    if A&C-B:
        v3.get_label_by_id('101').set_text('\n'.join(map(str,A&C-B)))
        v3.get_patch_by_id('101').set_alpha(0.3)
        v3.get_patch_by_id('101').set_facecolor('Green')
    if A&B&C:
        v3.get_label_by_id('111').set_text('\n'.join(map(str,A&B&C)))
        v3.get_patch_by_id('111').set_alpha(0.1)
        v3.get_patch_by_id('111').set_facecolor('Black')
    if B&C-A:
        v3.get_label_by_id('011').set_text('\n'.join(map(str,B&C-A)))
        v3.get_patch_by_id('011').set_alpha(0.3)
        v3.get_patch_by_id('011').set_facecolor('Blue')
    if C-B-A:
        v3.get_label_by_id('001').set_text('\n'.join(map(str,C-B-A)))
        v3.get_patch_by_id('001').set_alpha(0.6)
        v3.get_patch_by_id('001').set_facecolor('Cyan')


A={3, 4, 8, 12, 8, 2, 3, 4, 12, 3, 4}  # it contains repetitions
print(A)                               # Python automatically remove the repetitions and make them distinct

{2, 3, 4, 8, 12}


F={'durian','kiwi','apple', 'banana', 'cherry'}
print(F)   # Python automatially reorder the members of the set

{'banana', 'apple', 'durian', 'cherry', 'kiwi'}


A={1,3,6,8}
B={2,3,4,5}
C={5,8,9}

# now we want to find the universal set of the three sets above
U=set()
U=U.union(A,B,C)
U

{1, 2, 3, 4, 5, 6, 8, 9}


print('is A subset of U? ',A.issubset(U))
print('is B subset of U? ',B.issubset(U))
print('is C subset of U? ',C.issubset(U))
print('is U subset of U? ',U.issubset(U))
E=set() # empty set
print('is Empty set subset of U? ',E.issubset(U))

is A subset of U?  True
is B subset of U?  True
is C subset of U?  True
is U subset of U?  True
is Empty set subset of U?  True


C=set()
C

set()


C={}
C

{}


C={}
len(C)

0


Z={0}
len(Z)

1


F={'durian','kiwi','apple', 'banana', 'cherry'}
'apple' in F

True


F={'durian','kiwi','apple', 'banana', 'cherry'}
print('apple' not in F)
print('grape' not in F)

False
True


A={'a','b','c','d', 'e', 'f', 'g'}
len(A)

7


A={}
len(A)

0


C = {1, (2), ((4, 5), 6), ('red', 'blue', 'yellow')}
len(C)

4


F={'durian','kiwi','apple', 'banana', 'cherry'}
G={'apple', 'banana', 'cherry'}
print('is F a subset of G?', F.issubset(G))
print('is G a subset of F?', G.issubset(F))
print('is G a subset of G itself?',G.issubset(G))

is F a subset of G? False
is G a subset of F? True
is G a subset of G itself? True


X = {'a','e','i','o','u'}
Y = {'a', 'i', 'u'} 
Z= {'b','c','d','f','g'}
print('is Y a subset of X?',Y.issubset(X))
print('is Y a subset of Z?',Y.issubset(Z))
drawVenn3(X,Y,Z,labels=('X', 'Y', 'Z'))

is Y a subset of X? True
is Y a subset of Z? False


A = {3, 9}
B = {5, 9, 1, 3}
print('is A a subset of B?',A.issubset(B))
drawVenn2(A,B)

is A a subset of B? True


A = {3, 3, 3, 9}
B = {5, 9, 1, 3}
print('is A a subset of B?',A<=B)
drawVenn2(A,B)

is A a subset of B? True


A = {1, 2, 3}
B = {2, 3, 4}
print('is A a subset of B?',A.issubset(B))
drawVenn2(A,B)

is A a subset of B? False


A = {1, 2, 3}
B = {2, 3, 4}
print('is A not subset of B?',not A.issubset(B))

is A not subset of B? True


A = {3, 9, 5, 1}
B = {5, 9, 1, 3}
print('is A a subset of B?',A.issubset(B))
drawVenn2(A,B)

is A a subset of B? True


A = {3, 9, 5, 1}
B = {5, 9, 1, 3}
print('is A a proper subset of B?',A<=B)

is A a proper subset of B? True


A={5,9}
B={1,3,5,7,9}
C={1,2,3,4,5,6,7,8,9}
drawVenn3(A,B,C)


F={'durian','kiwi','apple', 'banana', 'cherry'}
G={'apple', 'banana', 'cherry'}

drawVenn2(F,G,labels=('F','G'))
print('is F superset of G?',F.issuperset(G))

is F superset of G? True


A = {1, 2, 3}
B = {2, 3, 4}
print('is A a superset of B?', A >= B)
drawVenn2(A,B)

is A a superset of B? False


A={5,9}
B={1,3,5,7,9}
print('is B a superset of A?', B >= A)
drawVenn2(A,B)

is B a superset of A? True


A = {1, 2, 3}
B = {2, 3, 1}
drawVenn2(A,B)
A==B

True


A = {1, 2, 3, 1, 2}
B = {2, 3, 1}
drawVenn2(A,B)
A==B

True


A.issubset(B) & B.issubset(A)

True


# https://www.delftstack.com/howto/python/powerset-python/
# produce list of list
def powerset(A):
    listsub = list(A)
    subsets = []
    for i in range(2**len(listsub)):
        subset = []
        for k in range(len(listsub)):            
            if i & 1<<k:
                subset.append(listsub[k])
        subsets.append(subset)        
    return subsets


# https://stackoverflow.com/questions/1482308/how-to-get-all-subsets-of-a-set-powerset
# produce set of tuple
from itertools import chain, combinations

def powerset(A):
    s = list(A)
    return set(chain.from_iterable(combinations(s, r) for r in range(len(s)+1)))


# https://stackoverflow.com/questions/41626379/python-power-set-of-a-list
# produce list of sets
def powerset(A):
    if not A: # Empty list -> empty set
        return [set()]

    r = []
    for y in A:
        sy = set((y,))
    for x in powerset(A - sy):
        if x not in r:
            r.extend([x, x|sy])
    return r


# https://stackoverflow.com/questions/68124242/generating-power-set-with-recursion-and-yield-statement-in-python
# produce list of list, the code is shortest
def powerset(A):
    ans = [[]]
    for n in A:
        ans += [a+[n] for a in ans]
    return ans


# A={'a','b','c','d', 'e', 'f', 'g'}
# A={'a','b','c','d'}
# A={'a','b','c'}
# A={'a','b'}
# A={'a'}
A={}
powerset(A)

[[]]


A={'x','y','z'}
powerset(A)

[[], ['y'], ['z'], ['y', 'z'], ['x'], ['y', 'x'], ['z', 'x'], ['y', 'z', 'x']]


X = {'red', 'blue', 'yellow'}
powerset(X)

[[],
 ['blue'],
 ['red'],
 ['blue', 'red'],
 ['yellow'],
 ['blue', 'yellow'],
 ['red', 'yellow'],
 ['blue', 'red', 'yellow']]


def cartesianProduct(A,B):
    return {(x,y) for x in A for y in B}


A = {'good', 'bad'}
B = {'student', 'prof'}
print(cartesianProduct(A,B))
print(cartesianProduct(B,A))

{('good', 'prof'), ('good', 'student'), ('bad', 'prof'), ('bad', 'student')}
{('student', 'bad'), ('prof', 'good'), ('student', 'good'), ('prof', 'bad')}


A = {'x', 'y'}
B = {'a', 'b', 'c'}
print(cartesianProduct(A,B))
print(cartesianProduct(B,A))

{('x', 'b'), ('y', 'a'), ('y', 'c'), ('x', 'a'), ('x', 'c'), ('y', 'b')}
{('a', 'x'), ('a', 'y'), ('c', 'x'), ('c', 'y'), ('b', 'x'), ('b', 'y')}


A = {'a', 'b', 'c'}
E = {}
print(cartesianProduct(A,E))
print(cartesianProduct(E,A))

set()
set()


x={'apple', 'banana', 'cherry', 'durian','kiwi'}
y={'durian','kiwi' ,'mango', 'orange','pineaple'}
x|y  # set union

{'apple', 'banana', 'cherry', 'durian', 'kiwi', 'mango', 'orange', 'pineaple'}


X = {'a','b','c','d','e'}
Y = {'d','e','f','g'}
print('X|Y =',X|Y)  
print('Y|X =',Y|X)  # show that union operation is commutative
drawVenn2(X,Y,labels=('X','Y'))

X|Y = {'g', 'd', 'b', 'c', 'f', 'e', 'a'}
Y|X = {'g', 'd', 'a', 'b', 'c', 'e', 'f'}


X = {'a','b','c','d','e'}
Y = {'d','e','f','g'}
print('X&Y =',X&Y)  
print('Y&X =',Y&X)  # show that intersection operation is commutative
drawVenn2(X,Y,labels=('X','Y'))

X&Y = {'e', 'd'}
Y&X = {'e', 'd'}


X = {1,2,3,4,}
Y = {6,7,8,9}
print(X.isdisjoint(Y))
drawVenn2(X,Y, labels=('X','Y'))

True


X = {'a','b','c','d','e'}
Y = {'d','e','f','g'}
print('X-Y =',X-Y)  
print('Y-X =',Y-X) # show that set difference operation is NOT commutative
drawVenn2(X,Y,labels=('X','Y'))

X-Y = {'c', 'a', 'b'}
Y-X = {'g', 'f'}


U = {1,2,3,4,5,6,7,8,9}
A = {3,4,5}
Ac= U-A
print('~A =',Ac)     # this is the complement
print(not A)  # be careful: this is not the set compplement but the logical value of the complement

~A = {1, 2, 6, 7, 8, 9}
False


E=set()
Ec=U-E
print('~E =',Ec)     # this is the complement
print(not E)         # be careful: this is not the set compplement but the logical value of the complement

~E = {1, 2, 3, 4, 5, 6, 7, 8, 9}
True


X = {'a','b','c','d','e'}
Y = {'d','e','f','g'}
drawVenn2(X,Y,labels=('X','Y'))
U = X.union(Y)
print('U =',U)                # this is the universal set
print('~X =',U-X)             # this is the complement of X
print('~Y =',U-Y)             # this is the complement of Y
print('~(X|Y) =',U-(X|Y))     # this is the complement of the union
print('~(X&Y) =',U-(X&Y))     # this is the complement of the intersection
print('~(X^Y) =',U-(X^Y))     # this is the complement of the symmetric difference

U = {'g', 'd', 'b', 'c', 'f', 'e', 'a'}
~X = {'g', 'f'}
~Y = {'c', 'a', 'b'}
~(X|Y) = set()
~(X&Y) = {'g', 'a', 'b', 'c', 'f'}
~(X^Y) = {'e', 'd'}


X = {'a','b','c','d','e'}
Y = {'d','e','f','g'}
print('X^Y =',X^Y)  
print('Y^X =',Y^X) 
drawVenn2(X,Y,labels=('X','Y'))
print('is Y^X = X^Y? ',Y^X==X^Y) # show that set symmetric difference operation is commutative

X^Y = {'g', 'a', 'b', 'c', 'f'}
Y^X = {'g', 'c', 'f', 'a', 'b'}
is Y^X = X^Y?  True


A = {'a','b','c','d','e'}
B = {'d','e','f','g','b'}
C = {'d','h','i','a','f'}
drawVenn3(A,B,C)
E = set()
U = E.union(A,B,C)

print('E=',E)
print('U=',U)

E= set()
U= {'g', 'i', 'd', 'b', 'c', 'f', 'e', 'h', 'a'}


print('A|E=',A|E)
print('A&U=',A&U)
print('is A|E==A ?', A|E==A)
print('is A&U==A ?',A&U==A)

A|E= {'c', 'e', 'd', 'a', 'b'}
A&U= {'d', 'b', 'c', 'e', 'a'}
is A|E==A ? True
is A&U==A ? True


print('A|U=',A|U)
print('A&E=',A&E)
print('is A|U==U ?', A|U==U)
print('is A&E==E ?',A&E==E)

A|U= {'g', 'i', 'd', 'b', 'c', 'f', 'e', 'h', 'a'}
A&E= set()
is A|U==U ? True
is A&E==E ? True


print('is A|A==A ?', A|A==A)
print('is A&A==A ?',A&A==A)

is A|A==A ? True
is A&A==A ? True


Ac = U-A       # A complement
print('A=',A)
print('~A=',Ac)
print('~(~A)=',U-Ac)
print('is ~(~A)==A ?', not (not A)==A)
U-Ac==A

A= {'d', 'b', 'c', 'e', 'a'}
~A= {'g', 'i', 'f', 'h'}
~(~A)= {'d', 'b', 'c', 'e', 'a'}
is ~(~A)==A ? True

True


print('is A|B==B|A ?', A|B==B|A)
print('is A&B==B&A ?', A&B==B&A)

is A|B==B|A ? True
is A&B==B&A ? True


print('is A|(B|C)==(A|B)|C ?', A|(B|C)==(A|B)|C)
print('is A&(B&C)==(A&B)&C ?', A&(B&C)==(A&B)&C)

is A|(B|C)==(A|B)|C ? True
is A&(B&C)==(A&B)&C ? True


print('is A&(B|C)==(A&B)|(A&C) ?', A&(B|C)==(A&B)|(A&C))
print('is A|(B&C)==(A|B)&(A|C) ?', A|(B&C)==(A|B)&(A|C))

is A&(B|C)==(A&B)|(A&C) ? True
is A|(B&C)==(A|B)&(A|C) ? True


print('is U-(A|B)==(U-A)&(U-B) ?', U-(A|B)==(U-A)&(U-B))
print('is U-(A&B)==(U-A)|(U-B) ?', U-(A&B)==(U-A)|(U-B))
print('is -(A|B)==(-A)&(-B) ?', not (A|B)==(not A)&(not B))  # this is logical, not set
print('is -(A&B)==(-A)|(-B) ?', not (A&B)==(not A)|(not B))  # this is logical, not set

is U-(A|B)==(U-A)&(U-B) ? True
is U-(A&B)==(U-A)|(U-B) ? True
is -(A|B)==(-A)&(-B) ? True
is -(A&B)==(-A)|(-B) ? True


print('is A|(A&B)==A ?', A|(A&B)==A)
print('is A&(A|B)==A ?', A&(A|B)==A)

is A|(A&B)==A ? True
is A&(A|B)==A ? True


print('U =',U)
print('A|Ac =',A|Ac)
print('A&Ac =',A&Ac)
print('is A|Ac==U?',A|Ac==U)
print('is A&Ac==E?',A&Ac==E)
print('is U-E==U?',U-E==U)
print('is U-U==E ?', U-U==E)

U = {'g', 'i', 'd', 'b', 'c', 'f', 'e', 'h', 'a'}
A|Ac = {'g', 'i', 'd', 'b', 'c', 'f', 'e', 'h', 'a'}
A&Ac = set()
is A|Ac==U? True
is A&Ac==E? True
is U-E==U? True
is U-U==E ? True

Set Theory in Python¶

Table of Contents¶

Preparation¶

What is Set?¶

Universal Set ¶

Empty Set ¶

Set Elements ¶

Cardinality of Sets ¶

Subset ¶

Not Subset ¶

Proper Subset ¶

Transitive Subsets ¶

Superset ¶

Equal Sets ¶

Power Set ¶

Relationship of Power set to Combination ¶

Cartesian Product ¶

Operations on Sets ¶

Union ¶

Intersection ¶

Disjoint ¶

Set Difference ¶

Complement ¶

Set Symmetric Difference ¶

Set Identities ¶

Identity laws ¶

Domination laws ¶

Idempotent laws ¶

Double Negation / Complementation / Involution law ¶

Commutative laws ¶

Associative laws ¶

Distributive laws ¶

De Morgan’s laws ¶

Absorption laws¶

Complement laws ¶

References ¶

Set Theory in Python¶

Table of Contents¶

Preparation¶

What is Set?¶

Universal Set¶

Empty Set¶

Set Elements¶

Cardinality of Sets¶

Subset¶

Not Subset¶

Proper Subset¶

Transitive Subsets¶

Superset¶

Equal Sets¶

Power Set¶

Relationship of Power set to Combination¶

Cartesian Product¶

Operations on Sets¶

Union¶

Intersection¶

Disjoint¶

Set Difference¶

Complement¶

Set Symmetric Difference¶

Set Identities¶

Identity laws ¶

Domination laws ¶

Idempotent laws ¶

Double Negation / Complementation / Involution law ¶

Commutative laws ¶

Associative laws ¶

Distributive laws ¶

De Morgan’s laws ¶

Absorption laws¶

Complement laws ¶

References ¶

Universal Set ¶

Empty Set ¶

Set Elements ¶

Cardinality of Sets ¶

Subset ¶

Not Subset ¶

Proper Subset ¶

Transitive Subsets ¶

Superset ¶

Equal Sets ¶

Power Set ¶

Relationship of Power set to Combination ¶

Cartesian Product ¶

Operations on Sets ¶

Union ¶

Intersection ¶

Disjoint ¶

Set Difference ¶

Complement ¶

Set Symmetric Difference ¶

Set Identities ¶